Background & Context

Sinh is the hyperbolic sine function, which is the hyperbolic analogue of the Sin circular function used throughout trigonometry. It is defined for real numbers by letting be twice the area between the axis and a ray through the origin intersecting the unit hyperbola . Sinh[α] then gives the vertical coordinate of the intersection point. Sinh may also be defined as , where is the base of the natural logarithm Log.

Sinh automatically evaluates to exact values when its argument is the (natural) logarithm of a rational number. When given exact numeric expressions as arguments, Sinh may be evaluated to arbitrary numeric precision. Other operations useful for manipulation of symbolic expressions involving Sinh include TrigToExp, TrigExpand, Simplify, and FullSimplify.

Sinh threads element-wise over lists and matrices. In contrast, MatrixFunction can be used to give the hyperbolic sine of a square matrix (i.e. the power series for the hyperbolic sine function with ordinary powers replaced by matrix powers) as opposed to the hyperbolic sines of the individual matrix elements.

Sinh[x] decreases exponentially as x approaches and increases exponentially as x approaches . Sinh satisfies an identity similar to the Pythagorean identity satisfied by Sin, namely . The definition of the hyperbolic sine function is extended to complex arguments by way of the identity . The hyperbolic sine function is entire, meaning it is complex differentiable at all finite points of the complex plane. Sinh[z] has series expansion about the origin.